A construction for regular-graph designs

TL;DR

This paper introduces a specific construction method for regular-graph designs with certain parameters, establishing existence conditions for small degrees and providing explicit designs for particular graph orders.

Contribution

It presents a new construction for regular-graph designs with λ=1 and block size δ+1, and proves sufficiency of conditions for δ=2,3 with some exceptions.

Findings

Necessary conditions for existence are sufficient for δ=2,3 with few exceptions.

Constructed designs for orders 105 and 117 for connected 4-regular graphs.

Established existence of regular-graph designs under specific parameters.

Abstract

A regular-graph design is a block design for which a pair $\{a,b\}$ of distinct points occurs in $\lambda+1$ or $\lambda$ blocks depending on whether $\{a,b\}$ is or is not an edge of a given $\delta$-regular graph. Our paper describes a specific construction for regular-graph designs with $\lambda = 1$ and block size $\delta + 1$. We show that for $\delta \in \{2,3\}$, certain necessary conditions for the existence of such a design with $n$ points are sufficient, with two exceptions in each case and two possible exceptions when $\delta = 3$. We also construct designs of orders 105 and 117 for connected 4-regular graphs.